Bayesian Neural Network
We borrow this tutorial from the official Turing Docs. We will show how the explicit parameterization of Lux enables first-class composability with packages which expect flattened out parameter vectors.
We will use Turing.jl with Lux.jl to implement implementing a classification algorithm. Lets start by importing the relevant libraries
# Import libraries.
using Lux
using Turing, Plots, Random, ReverseDiff, NNlib, Functors
# Hide sampling progress.
Turing.setprogress!(false);
# Use reverse_diff due to the number of parameters in neural networks.
Turing.setadbackend(:reversediff):reversediffGenerating data
Our goal here is to use a Bayesian neural network to classify points in an artificial dataset. The code below generates data points arranged in a box-like pattern and displays a graph of the dataset we'll be working with.
# Number of points to generate.
N = 80
M = round(Int, N / 4)
rng = Random.default_rng()
Random.seed!(rng, 1234)
# Generate artificial data.
x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
xt1s = Array([[x1s[i] + 0.5f0; x2s[i] + 0.5f0] for i in 1:M])
x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
append!(xt1s, Array([[x1s[i] - 5.0f0; x2s[i] - 5.0f0] for i in 1:M]))
x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
xt0s = Array([[x1s[i] + 0.5f0; x2s[i] - 5.0f0] for i in 1:M])
x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
append!(xt0s, Array([[x1s[i] - 5.0f0; x2s[i] + 0.5f0] for i in 1:M]))
# Store all the data for later.
xs = [xt1s; xt0s]
ts = [ones(2 * M); zeros(2 * M)]
# Plot data points.
function plot_data()
x1 = first.(xt1s)
y1 = last.(xt1s)
x2 = first.(xt0s)
y2 = last.(xt0s)
plt = Plots.scatter(x1, y1; color="red", clim=(0, 1))
Plots.scatter!(plt, x2, y2; color="blue", clim=(0, 1))
return plt
end
plot_data()Building the Neural Network
The next step is to define a feedforward neural network where we express our parameters as distributions, and not single points as with traditional neural networks. For this we will use Dense to define liner layers and compose them via Chain, both are neural network primitives from Lux. The network nn we will creat will have two hidden layers with tanh activations and one output layer with sigmoid activation, as shown below.
The nn is an instance that acts as a function and can take data, parameters and current state as inputs and output predictions. We will define distributions on the neural network parameters.
# Construct a neural network using Lux
nn = Chain(Dense(2, 3, tanh), Dense(3, 2, tanh), Dense(2, 1, sigmoid))
# Initialize the model weights and state
ps, st = Lux.setup(rng, nn)
Lux.parameterlength(nn) # number of paraemters in NN20The probabilistic model specification below creates a parameters variable, which has IID normal variables. The parameters represents all parameters of our neural net (weights and biases).
# Create a regularization term and a Gaussian prior variance term.
alpha = 0.09
sig = sqrt(1.0 / alpha)3.3333333333333335Construct named tuple from a sampled parameter vector. We could also use ComponentArrays here and simply broadcast to avoid doing this. But let's do it this way to avoid dependencies.
function vector_to_parameters(ps_new::AbstractVector, ps::NamedTuple)
@assert length(ps_new) == Lux.parameterlength(ps)
i = 1
function get_ps(x)
z = reshape(view(ps_new, i:(i + length(x) - 1)), size(x))
i += length(x)
return z
end
return fmap(get_ps, ps)
end
# Specify the probabilistic model.
@model function bayes_nn(xs, ts)
global st
# Sample the parameters
nparameters = Lux.parameterlength(nn)
parameters ~ MvNormal(zeros(nparameters), sig .* ones(nparameters))
# Forward NN to make predictions
preds, st = nn(xs, vector_to_parameters(parameters, ps), st)
# Observe each prediction.
for i in 1:length(ts)
ts[i] ~ Bernoulli(preds[i])
end
endbayes_nn (generic function with 2 methods)Inference can now be performed by calling sample. We use the HMC sampler here.
# Perform inference.
N = 5000
ch = sample(bayes_nn(hcat(xs...), ts), HMC(0.05, 4), N)Chains MCMC chain (5000×29×1 Array{Float64, 3}):
Iterations = 1:1:5000
Number of chains = 1
Samples per chain = 5000
Wall duration = 97.94 seconds
Compute duration = 97.94 seconds
parameters = parameters[1], parameters[2], parameters[3], parameters[4], parameters[5], parameters[6], parameters[7], parameters[8], parameters[9], parameters[10], parameters[11], parameters[12], parameters[13], parameters[14], parameters[15], parameters[16], parameters[17], parameters[18], parameters[19], parameters[20]
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, step_size, nom_step_size
Summary Statistics
parameters mean std naive_se mcse ess rhat ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
parameters[1] 1.4675 2.3083 0.0326 0.2693 11.7281 2.0767 ⋯
parameters[2] 5.5552 2.7594 0.0390 0.3231 12.4936 1.1899 ⋯
parameters[3] 0.0273 0.7531 0.0107 0.0774 18.6028 1.1024 ⋯
parameters[4] -1.8129 1.6025 0.0227 0.1796 15.6117 1.0448 ⋯
parameters[5] 0.7135 1.3562 0.0192 0.1557 13.5410 1.1842 ⋯
parameters[6] 4.7089 1.7599 0.0249 0.1976 18.8688 1.0080 ⋯
parameters[7] -5.0105 3.4342 0.0486 0.4005 12.7796 1.0072 ⋯
parameters[8] 0.1951 2.3736 0.0336 0.2763 12.0470 1.4253 ⋯
parameters[9] 0.8814 1.6137 0.0228 0.1798 19.0521 1.0278 ⋯
parameters[10] -0.9196 4.1529 0.0587 0.4896 10.7452 2.1730 ⋯
parameters[11] -0.0995 2.6126 0.0369 0.3022 12.2717 1.2131 ⋯
parameters[12] -2.0922 3.0141 0.0426 0.3552 11.4220 1.6551 ⋯
parameters[13] 4.3457 1.7685 0.0250 0.2022 14.9620 1.0385 ⋯
parameters[14] -2.9048 1.6752 0.0237 0.1877 15.8258 1.0980 ⋯
parameters[15] 2.1551 1.8145 0.0257 0.2082 13.7275 1.2178 ⋯
parameters[16] -3.2932 1.4081 0.0199 0.1561 19.8148 1.0165 ⋯
parameters[17] -3.4255 2.6896 0.0380 0.3158 12.6164 1.2372 ⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱
1 column and 3 rows omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
parameters[1] -2.8737 -0.2060 1.2397 3.2028 6.3763
parameters[2] 0.7006 3.3812 5.8509 7.5387 11.0436
parameters[3] -2.0910 -0.2111 0.1039 0.4571 1.0501
parameters[4] -5.8984 -2.5130 -1.6280 -0.7613 0.8443
parameters[5] -0.7348 -0.0564 0.4240 0.9659 5.2736
parameters[6] 1.6909 3.4942 4.5379 5.9346 8.2718
parameters[7] -11.1144 -7.4680 -5.2027 -2.8930 1.9571
parameters[8] -4.3334 -1.5037 0.2996 2.0048 4.4496
parameters[9] -1.9076 -0.1331 0.6428 1.7236 4.4656
parameters[10] -8.4372 -4.2836 0.1960 2.2765 5.9196
parameters[11] -5.1766 -1.7353 0.2017 1.6669 4.5451
parameters[12] -6.9702 -4.6678 -2.4274 0.6276 3.0518
parameters[13] 1.2283 3.0997 4.1223 5.5624 8.0027
parameters[14] -6.2349 -4.0604 -2.9481 -1.8547 0.5993
parameters[15] -1.9406 1.1370 2.4385 3.4656 5.0130
parameters[16] -5.6563 -4.2682 -3.5360 -2.4030 0.0173
parameters[17] -9.2556 -4.9700 -3.1804 -1.7896 1.5551
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
3 rows omitted
Now we extract the parameter samples from the sampled chain as theta (this is of size 5000 x 20 where 5000 is the number of iterations and 20 is the number of parameters). We'll use these primarily to determine how good our model's classifier is.
# Extract all weight and bias parameters.
theta = MCMCChains.group(ch, :parameters).value;Prediction Visualization
# A helper to run the nn through data `x` using parameters `theta`
nn_forward(x, theta) = nn(x, vector_to_parameters(theta, ps), st)[1]
# Plot the data we have.
plot_data()
# Find the index that provided the highest log posterior in the chain.
_, i = findmax(ch[:lp])
# Extract the max row value from i.
i = i.I[1]
# Plot the posterior distribution with a contour plot
x1_range = collect(range(-6; stop=6, length=25))
x2_range = collect(range(-6; stop=6, length=25))
Z = [nn_forward([x1, x2], theta[i, :])[1] for x1 in x1_range, x2 in x2_range]
contour!(x1_range, x2_range, Z)The contour plot above shows that the MAP method is not too bad at classifying our data. Now we can visualize our predictions.
\[p(\tilde{x} | X, \alpha) = \int_{\theta} p(\tilde{x} | \theta) p(\theta | X, \alpha) \approx \sum_{\theta \sim p(\theta | X, \alpha)}f_{\theta}(\tilde{x})\]
The nn_predict function takes the average predicted value from a network parameterized by weights drawn from the MCMC chain.
# Return the average predicted value across multiple weights.
function nn_predict(x, theta, num)
return mean([nn_forward(x, view(theta, i, :))[1] for i in 1:10:num])
endnn_predict (generic function with 1 method)Next, we use the nn_predict function to predict the value at a sample of points where the x1 and x2 coordinates range between -6 and 6. As we can see below, we still have a satisfactory fit to our data, and more importantly, we can also see where the neural network is uncertain about its predictions much easier–-those regions between cluster boundaries.
Plot the average prediction.
plot_data()
n_end = 1500
x1_range = collect(range(-6; stop=6, length=25))
x2_range = collect(range(-6; stop=6, length=25))
Z = [nn_predict([x1, x2], theta, n_end)[1] for x1 in x1_range, x2 in x2_range]
contour!(x1_range, x2_range, Z)Suppose we are interested in how the predictive power of our Bayesian neural network evolved between samples. In that case, the following graph displays an animation of the contour plot generated from the network weights in samples 1 to 1,000.
# Number of iterations to plot.
n_end = 1000
anim = @gif for i in 1:n_end
plot_data()
Z = [nn_forward([x1, x2], theta[i, :])[1] for x1 in x1_range, x2 in x2_range]
contour!(x1_range, x2_range, Z; title="Iteration $i", clim=(0, 1))
end every 5
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